Complex, high-dimensional data like multi-platform genomics and imaging data can be used to discover biomarkers providing insight into cancer etiology, natural history, prognosis, and prediction of response to therapy. Existing analytical methods are not adequate, however, as most either ignore important structure in the data or limit analysis to simple summaries that do not use all of the information in the data. This research will develop a general suite of flexible, automated, novel Bayesian methods for performing regression analyses on complex, high dimensional functional data to discover biomarkers using models that account for their intricate structure, yield inference that adjusts fo multiple testing, and are scalable to high-dimensional settings. While generally applicable, these methods will be developed in the context of two studies conducted by our collaborators to discover early genomic and epigenetic events in the natural history of bladder cancer and neuroimaging biomarkers associated with and predicting smoking cessation success. Specific Aim 1: Modeling multi-platform genomic data as functions, we will develop methods for functional response regression for spatially correlated genomics data on a lattice generated by a novel bladder cancer model developed by our co-I Czerniak. We will apply these methods to identify genomic and epigenetic changes in bladder cancer and determine when first observed in the disease's natural history, revealing early aberrations that are potential disease drivers. We will develop inferential strategies to perform genome-level tests and then ag genomic regions while adjusting for multiplicity. Specific Aim 2: We will develop functional regression approaches for event-related potentials (ERPs) from a randomized smoking cessation trial conducted by our co-Is Cinciripini and Versace to test whether different emotional stimuli evoke differential neurological response, determine whether these effects vary between individuals successful or unsuccessful in their smoking cessation attempt, and assess whether ERPs are independent predictors of success. Our methods will flexibly capture inter-electrode correlation via spatial functional processes or tensor basis functions, and capture intra-electrode correlation using basis function modeling, with strategies to determine which basis is best for ERPs. Specific Aim 3: We will develop functional regression approaches for fMRI data from our smoking cessation trial, first at the subject level to identify brain regions differentially activaed by different visual stimuli, and then introducing a strategy to scale our approach up to group-level analyses to characterize population-level neurological differences, relate them to cessation success, and assess their predictive ability relative to ERP and standard demographic, psychometric, and genetic predictors. Our models for longitudinally correlated volumetric data will capture intra-volume correlation through basis functional modeling, introducing a novel hybrid basis function modeling strategy that captures within-brain correlation in a manner that accounts for known anatomy, spatial proximity, and distant correlations induced by functional connectivity. Specific Aim 4: We will integrate these new methods into a general suite of Bayesian methods for spatially and longitudinally correlated functional response regression, discrimination, and inference for complex, high-dimensional functions along with freely available, automated, scalable software that can be broadly applied.